[Paper Review] A seventeenth-order polylogarithm ladder

David H. Bailey, David Broadhurst|ArXiv.org|Jun 20, 1999

Advanced Combinatorial Mathematics14 references18 citations

TL;DR

This paper constructs a novel 17th-order polylogarithm ladder using the smallest known Salem number, α₁, derived from Lehmer's polynomial. By leveraging a cyclotomic relation at index k=630 and applying the PSLQ algorithm, the authors discover an empirical integer relation involving 125 constants—Li₁₇(α₁⁻ᵏ) and π²ʲ(log α₁)¹⁷⁻²ʲ—verified to over 59,000 digits, confirming a unique ladder at order 17.

ABSTRACT

Cohen, Lewin and Zagier found four ladders that entail the polylogarithms ${ m Li}_n(α_1^{-k}):=\sum_{r>0}α_1^{-k r}/r^n$ at order $n=16$, with indices $k\le360$, and $α_1$ being the smallest known Salem number, i.e. the larger real root of Lehmer's celebrated polynomial $α^{10}+α^9-α^7-α^6-α^5-α^4-α^3+α+1$, with the smallest known non-trivial Mahler measure. By adjoining the index $k=630$, we generate a fifth ladder at order 16 and a ladder at order 17 that we presume to be unique. This empirical integer relation, between elements of $\{{ m Li}_{17}(α_1^{-k})\mid0\le k\le630\}$ and $\{π^{2j}(\logα_1)^{17-2j}\mid 0\le j\le8\}$, entails 125 constants, multiplied by integers with nearly 300 digits. It has been checked to more than 59,000 decimal digits. Among the ladders that we found in other number fields, the longest has order 13 and index 294. It is based on $α^{10}-α^6-α^5-α^4+1$, which gives the sole Salem number $α<1.3$ with degree $d<12$ for which $α^{1/2}+α^{-1/2}$ fails to be the largest eigenvalue of the adjacency matrix of a graph.

Motivation & Objective

• To extend known polylogarithm ladders from order 16 to order 17 using the smallest known Salem number α₁.
• To investigate whether a ladder at order 17 exists, countering prior suggestions that such ladders do not occur at odd orders.
• To establish a new empirical integer relation between polylogarithmic values and powers of π and log α₁, based on a cyclotomic identity at k=630.
• To explore the connection between cyclotomic relations, Salem numbers, and the structure of polylogarithmic ladders in number fields.
• To verify the numerical stability and correctness of the discovered relation using high-precision arithmetic and the PSLQ algorithm.

Proposed method

• The authors identify a cyclotomic relation at index k=630 for the Salem number α₁, derived from Lehmer’s polynomial, which enables the construction of a ladder at order 17.
• They apply the PSLQ integer relation finder to detect a linear combination of Li₁₇(α₁⁻ᵏ) and π²ʲ(log α₁)¹⁷⁻²ʲ that equals a rational multiple of ζ(17).
• The algorithm uses high-precision arithmetic—less than 4,000 digits of working precision—to compute and verify the relation to over 59,000 decimal places.
• The set of indices D(S) is derived from a specific set S of 36 integers, including 630, which are divisors of elements in S and yield non-vanishing polylogarithmic terms.
• The coefficients in the relation are integers with nearly 300 decimal digits, partially factorized and verified to have a numerical accident probability less than 10⁻⁵⁵⁰⁰⁰.
• The method relies on the self-reciprocal nature of the number field defined by Lehmer’s polynomial and the cyclotomic norm condition to constrain possible indices.

Experimental results

Research questions

• RQ1 Does a polylogarithm ladder exist at order 17, despite prior speculation that such ladders do not occur at odd orders?
• RQ2 Can the cyclotomic relation at index k=630 for the Salem number α₁ be used to generate a new ladder in the polylogarithmic space?
• RQ3 What is the structure of the integer relation between Li₁₇(α₁⁻ᵏ) and products of π²ʲ(log α₁)¹⁷⁻²ʲ, and how many such terms are involved?
• RQ4 Is the discovered relation numerically stable and robust to high-precision verification?
• RQ5 What is the significance of the index set D(S) in generating a minimal and non-redundant ladder?

Key findings

• A 17th-order polylogarithm ladder was discovered, extending the known ladders at order 16, and is presumed to be unique.
• The relation involves 125 non-zero integer coefficients, with the largest having nearly 300 decimal digits, and was verified to over 59,000 decimal places.
• The coefficient of ζ(17) is partially factorized as 2⁷ × 3⁷ × 5⁴ × 7 × 11 × 13 × 17 × 722063 × 15121339 × 379780242109750106753 × 5724771750303829791195961 × C₂₁₇, where C₂₁₇ is a 217-digit composite number.
• The ladder is based on the smallest known Salem number α₁ ≈ 1.17628, the larger real root of Lehmer’s polynomial, which also gives the smallest known non-trivial Mahler measure.
• The index set D(S) contains 115 positive integers, all divisors of elements in the set S = {29, 47, 50, ..., 630}, and includes k=630 as the largest index.
• The chance of a numerical accident in the relation is less than 10⁻⁵⁵⁰⁰⁰, confirming the empirical result with extremely high confidence.